Leech lattice

In mathematics, the Leech lattice is a lattice Λ in R²⁴ discovered John Leech (Canad. J. Math. 16 (1964), 657--682). It is the unique lattice with the following list of properties:

• It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1.
• It is even; i.e., the square of the length of any vector in Λ is an even integer.
• The shortest length of any non-zero vector in Λ is 2.

The last condition means that unit balls centered at the points of Λ do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls which can simultaneously touch a single unit ball (compare with 6 in dimension 2, as the maximum number of pennies which can touch a central penny). It seems to be expected that this configuration also gives the densest packing of balls in 24-dimensional space, but this is still open.

The Leech lattice can be explicitly constructed as the set of vectors of the form 2^−3/2(a₁, a₂, ..., a₂₄) where the a_i are integers such that

and the set of coordinates i such that a_i belongs to any fixed residue class (mod 4) is a word in the binary Golay code.

The Leech lattice is highly symmetrical. Its automorphism group is the double cover of the Conway group Co₁; its order is approximately 8.3(10)¹⁸.

See:

• Conway, J. H.; Sloane, N. J. A. (1999). Sphere packings, lattices and groups. (3rd ed.) With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften, 290. New York: Springer-Verlag. ISBN 0-387-98585-9.